Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis

Abstract

Currently, among the open (unsolved) problems in number theory is the following: it is unknown what is the natural density of the sequence of Mersenne numbers in the set of natural numbers. In the paper, using methods of nonstandard mathematical analysis, we obtain the following equation: the natural density of Mersenne numbers (some infinitesimal value $e$) multiplied by the sum of the reciprocals of odd numbers (the infinitely large value ω = 1 + 1/3 + 1/5 + ...) is equal to 1, or the equality $e$ = 1/ω is true. In nonstandard analysis, the resulting infinitesimal numbers $e$ and 1/ω are considered equivalent. We obtained this result by working with a two-dimensional matrix of non-negative integers, where odd numbers are separated from even ones by the Pepis-Kalmar pairing function.